Properties

Label 888.2.bo.c.601.4
Level $888$
Weight $2$
Character 888.601
Analytic conductor $7.091$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [888,2,Mod(49,888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(888, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("888.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bo (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 601.4
Character \(\chi\) \(=\) 888.601
Dual form 888.2.bo.c.625.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.173648 - 0.984808i) q^{3} +(2.85598 + 2.39645i) q^{5} +(-0.585166 - 0.491013i) q^{7} +(-0.939693 - 0.342020i) q^{9} +O(q^{10})\) \(q+(0.173648 - 0.984808i) q^{3} +(2.85598 + 2.39645i) q^{5} +(-0.585166 - 0.491013i) q^{7} +(-0.939693 - 0.342020i) q^{9} +(1.49968 + 2.59752i) q^{11} +(0.665439 - 0.242200i) q^{13} +(2.85598 - 2.39645i) q^{15} +(2.13726 + 0.777899i) q^{17} +(0.592912 - 3.36257i) q^{19} +(-0.585166 + 0.491013i) q^{21} +(-0.381940 + 0.661540i) q^{23} +(1.54540 + 8.76441i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(4.38934 + 7.60256i) q^{29} -4.76785 q^{31} +(2.81847 - 1.02584i) q^{33} +(-0.494535 - 2.80465i) q^{35} +(5.04993 + 3.39091i) q^{37} +(-0.122968 - 0.697387i) q^{39} +(-4.61617 + 1.68015i) q^{41} +1.52668 q^{43} +(-1.86411 - 3.22873i) q^{45} +(5.60329 - 9.70517i) q^{47} +(-1.11421 - 6.31901i) q^{49} +(1.13721 - 1.96971i) q^{51} +(2.09133 - 1.75483i) q^{53} +(-1.94178 + 11.0124i) q^{55} +(-3.20853 - 1.16781i) q^{57} +(-2.71806 + 2.28073i) q^{59} +(0.305639 - 0.111243i) q^{61} +(0.381940 + 0.661540i) q^{63} +(2.48090 + 0.902974i) q^{65} +(3.97888 + 3.33868i) q^{67} +(0.585166 + 0.491013i) q^{69} +(1.46222 - 8.29267i) q^{71} -9.28253 q^{73} +8.89961 q^{75} +(0.397853 - 2.25634i) q^{77} +(6.79079 + 5.69815i) q^{79} +(0.766044 + 0.642788i) q^{81} +(-1.02771 - 0.374056i) q^{83} +(4.23977 + 7.34350i) q^{85} +(8.24927 - 3.00249i) q^{87} +(-0.125961 + 0.105694i) q^{89} +(-0.508315 - 0.185012i) q^{91} +(-0.827928 + 4.69541i) q^{93} +(9.75159 - 8.18256i) q^{95} +(-4.40886 + 7.63637i) q^{97} +(-0.520832 - 2.95379i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{5} + 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 3 q^{5} + 15 q^{7} + 12 q^{13} + 3 q^{15} + 3 q^{17} + 9 q^{19} + 15 q^{21} + 27 q^{25} - 12 q^{27} - 6 q^{29} - 30 q^{31} + 9 q^{33} + 15 q^{35} + 9 q^{37} + 3 q^{39} + 15 q^{41} - 54 q^{43} + 6 q^{45} - 12 q^{47} + 27 q^{49} + 18 q^{51} + 39 q^{53} - 6 q^{55} - 3 q^{59} + 12 q^{61} + 36 q^{65} + 48 q^{67} - 15 q^{69} + 33 q^{71} - 48 q^{73} + 60 q^{75} + 36 q^{77} + 18 q^{79} - 42 q^{83} + 15 q^{87} + 36 q^{89} - 36 q^{91} - 18 q^{93} + 27 q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{9}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.173648 0.984808i 0.100256 0.568579i
\(4\) 0 0
\(5\) 2.85598 + 2.39645i 1.27723 + 1.07173i 0.993619 + 0.112788i \(0.0359780\pi\)
0.283614 + 0.958938i \(0.408466\pi\)
\(6\) 0 0
\(7\) −0.585166 0.491013i −0.221172 0.185585i 0.525469 0.850813i \(-0.323890\pi\)
−0.746641 + 0.665228i \(0.768334\pi\)
\(8\) 0 0
\(9\) −0.939693 0.342020i −0.313231 0.114007i
\(10\) 0 0
\(11\) 1.49968 + 2.59752i 0.452170 + 0.783181i 0.998521 0.0543757i \(-0.0173169\pi\)
−0.546351 + 0.837556i \(0.683984\pi\)
\(12\) 0 0
\(13\) 0.665439 0.242200i 0.184559 0.0671741i −0.248087 0.968738i \(-0.579802\pi\)
0.432647 + 0.901564i \(0.357580\pi\)
\(14\) 0 0
\(15\) 2.85598 2.39645i 0.737411 0.618761i
\(16\) 0 0
\(17\) 2.13726 + 0.777899i 0.518361 + 0.188668i 0.587934 0.808909i \(-0.299941\pi\)
−0.0695728 + 0.997577i \(0.522164\pi\)
\(18\) 0 0
\(19\) 0.592912 3.36257i 0.136023 0.771427i −0.838118 0.545489i \(-0.816344\pi\)
0.974142 0.225938i \(-0.0725448\pi\)
\(20\) 0 0
\(21\) −0.585166 + 0.491013i −0.127694 + 0.107148i
\(22\) 0 0
\(23\) −0.381940 + 0.661540i −0.0796400 + 0.137941i −0.903095 0.429442i \(-0.858710\pi\)
0.823455 + 0.567382i \(0.192044\pi\)
\(24\) 0 0
\(25\) 1.54540 + 8.76441i 0.309080 + 1.75288i
\(26\) 0 0
\(27\) −0.500000 + 0.866025i −0.0962250 + 0.166667i
\(28\) 0 0
\(29\) 4.38934 + 7.60256i 0.815080 + 1.41176i 0.909270 + 0.416206i \(0.136641\pi\)
−0.0941897 + 0.995554i \(0.530026\pi\)
\(30\) 0 0
\(31\) −4.76785 −0.856331 −0.428165 0.903700i \(-0.640840\pi\)
−0.428165 + 0.903700i \(0.640840\pi\)
\(32\) 0 0
\(33\) 2.81847 1.02584i 0.490633 0.178576i
\(34\) 0 0
\(35\) −0.494535 2.80465i −0.0835916 0.474072i
\(36\) 0 0
\(37\) 5.04993 + 3.39091i 0.830203 + 0.557461i
\(38\) 0 0
\(39\) −0.122968 0.697387i −0.0196907 0.111671i
\(40\) 0 0
\(41\) −4.61617 + 1.68015i −0.720925 + 0.262395i −0.676318 0.736610i \(-0.736426\pi\)
−0.0446063 + 0.999005i \(0.514203\pi\)
\(42\) 0 0
\(43\) 1.52668 0.232816 0.116408 0.993201i \(-0.462862\pi\)
0.116408 + 0.993201i \(0.462862\pi\)
\(44\) 0 0
\(45\) −1.86411 3.22873i −0.277885 0.481311i
\(46\) 0 0
\(47\) 5.60329 9.70517i 0.817323 1.41565i −0.0903245 0.995912i \(-0.528790\pi\)
0.907648 0.419733i \(-0.137876\pi\)
\(48\) 0 0
\(49\) −1.11421 6.31901i −0.159173 0.902715i
\(50\) 0 0
\(51\) 1.13721 1.96971i 0.159242 0.275814i
\(52\) 0 0
\(53\) 2.09133 1.75483i 0.287266 0.241045i −0.487755 0.872981i \(-0.662184\pi\)
0.775021 + 0.631936i \(0.217739\pi\)
\(54\) 0 0
\(55\) −1.94178 + 11.0124i −0.261829 + 1.48491i
\(56\) 0 0
\(57\) −3.20853 1.16781i −0.424980 0.154680i
\(58\) 0 0
\(59\) −2.71806 + 2.28073i −0.353862 + 0.296925i −0.802339 0.596869i \(-0.796411\pi\)
0.448477 + 0.893794i \(0.351967\pi\)
\(60\) 0 0
\(61\) 0.305639 0.111243i 0.0391330 0.0142433i −0.322379 0.946611i \(-0.604483\pi\)
0.361513 + 0.932367i \(0.382260\pi\)
\(62\) 0 0
\(63\) 0.381940 + 0.661540i 0.0481199 + 0.0833462i
\(64\) 0 0
\(65\) 2.48090 + 0.902974i 0.307718 + 0.112000i
\(66\) 0 0
\(67\) 3.97888 + 3.33868i 0.486098 + 0.407884i 0.852625 0.522523i \(-0.175009\pi\)
−0.366528 + 0.930407i \(0.619453\pi\)
\(68\) 0 0
\(69\) 0.585166 + 0.491013i 0.0704457 + 0.0591110i
\(70\) 0 0
\(71\) 1.46222 8.29267i 0.173534 0.984159i −0.766289 0.642496i \(-0.777899\pi\)
0.939823 0.341663i \(-0.110990\pi\)
\(72\) 0 0
\(73\) −9.28253 −1.08644 −0.543219 0.839591i \(-0.682795\pi\)
−0.543219 + 0.839591i \(0.682795\pi\)
\(74\) 0 0
\(75\) 8.89961 1.02764
\(76\) 0 0
\(77\) 0.397853 2.25634i 0.0453396 0.257134i
\(78\) 0 0
\(79\) 6.79079 + 5.69815i 0.764023 + 0.641091i 0.939171 0.343451i \(-0.111596\pi\)
−0.175148 + 0.984542i \(0.556040\pi\)
\(80\) 0 0
\(81\) 0.766044 + 0.642788i 0.0851160 + 0.0714208i
\(82\) 0 0
\(83\) −1.02771 0.374056i −0.112806 0.0410580i 0.285001 0.958527i \(-0.408006\pi\)
−0.397806 + 0.917469i \(0.630228\pi\)
\(84\) 0 0
\(85\) 4.23977 + 7.34350i 0.459868 + 0.796515i
\(86\) 0 0
\(87\) 8.24927 3.00249i 0.884414 0.321900i
\(88\) 0 0
\(89\) −0.125961 + 0.105694i −0.0133519 + 0.0112036i −0.649439 0.760414i \(-0.724996\pi\)
0.636087 + 0.771617i \(0.280552\pi\)
\(90\) 0 0
\(91\) −0.508315 0.185012i −0.0532859 0.0193945i
\(92\) 0 0
\(93\) −0.827928 + 4.69541i −0.0858522 + 0.486892i
\(94\) 0 0
\(95\) 9.75159 8.18256i 1.00049 0.839513i
\(96\) 0 0
\(97\) −4.40886 + 7.63637i −0.447652 + 0.775355i −0.998233 0.0594263i \(-0.981073\pi\)
0.550581 + 0.834782i \(0.314406\pi\)
\(98\) 0 0
\(99\) −0.520832 2.95379i −0.0523456 0.296867i
\(100\) 0 0
\(101\) 9.05085 15.6765i 0.900593 1.55987i 0.0738664 0.997268i \(-0.476466\pi\)
0.826726 0.562604i \(-0.190201\pi\)
\(102\) 0 0
\(103\) −0.906253 1.56968i −0.0892957 0.154665i 0.817918 0.575335i \(-0.195128\pi\)
−0.907214 + 0.420670i \(0.861795\pi\)
\(104\) 0 0
\(105\) −2.84791 −0.277928
\(106\) 0 0
\(107\) 4.19247 1.52594i 0.405302 0.147518i −0.131321 0.991340i \(-0.541922\pi\)
0.536623 + 0.843822i \(0.319700\pi\)
\(108\) 0 0
\(109\) −0.549855 3.11838i −0.0526666 0.298687i 0.947085 0.320983i \(-0.104013\pi\)
−0.999751 + 0.0222964i \(0.992902\pi\)
\(110\) 0 0
\(111\) 4.21630 4.38438i 0.400194 0.416147i
\(112\) 0 0
\(113\) −2.35688 13.3666i −0.221717 1.25742i −0.868863 0.495053i \(-0.835149\pi\)
0.647146 0.762366i \(-0.275963\pi\)
\(114\) 0 0
\(115\) −2.67616 + 0.974043i −0.249553 + 0.0908300i
\(116\) 0 0
\(117\) −0.708145 −0.0654680
\(118\) 0 0
\(119\) −0.868694 1.50462i −0.0796330 0.137928i
\(120\) 0 0
\(121\) 1.00194 1.73541i 0.0910854 0.157765i
\(122\) 0 0
\(123\) 0.853034 + 4.83779i 0.0769155 + 0.436209i
\(124\) 0 0
\(125\) −7.26931 + 12.5908i −0.650187 + 1.12616i
\(126\) 0 0
\(127\) −9.79685 + 8.22053i −0.869330 + 0.729454i −0.963957 0.266059i \(-0.914279\pi\)
0.0946273 + 0.995513i \(0.469834\pi\)
\(128\) 0 0
\(129\) 0.265105 1.50348i 0.0233412 0.132374i
\(130\) 0 0
\(131\) −17.3547 6.31660i −1.51629 0.551884i −0.556070 0.831136i \(-0.687691\pi\)
−0.960218 + 0.279252i \(0.909913\pi\)
\(132\) 0 0
\(133\) −1.99802 + 1.67654i −0.173250 + 0.145374i
\(134\) 0 0
\(135\) −3.50338 + 1.27513i −0.301523 + 0.109745i
\(136\) 0 0
\(137\) 8.18469 + 14.1763i 0.699266 + 1.21116i 0.968721 + 0.248151i \(0.0798229\pi\)
−0.269456 + 0.963013i \(0.586844\pi\)
\(138\) 0 0
\(139\) 2.87169 + 1.04521i 0.243574 + 0.0886537i 0.460922 0.887440i \(-0.347519\pi\)
−0.217349 + 0.976094i \(0.569741\pi\)
\(140\) 0 0
\(141\) −8.58473 7.20344i −0.722965 0.606639i
\(142\) 0 0
\(143\) 1.62706 + 1.36527i 0.136062 + 0.114169i
\(144\) 0 0
\(145\) −5.68331 + 32.2316i −0.471973 + 2.67669i
\(146\) 0 0
\(147\) −6.41649 −0.529223
\(148\) 0 0
\(149\) −7.99238 −0.654761 −0.327380 0.944893i \(-0.606166\pi\)
−0.327380 + 0.944893i \(0.606166\pi\)
\(150\) 0 0
\(151\) 2.97374 16.8649i 0.242000 1.37245i −0.585359 0.810774i \(-0.699046\pi\)
0.827359 0.561674i \(-0.189842\pi\)
\(152\) 0 0
\(153\) −1.74231 1.46197i −0.140857 0.118193i
\(154\) 0 0
\(155\) −13.6169 11.4259i −1.09373 0.917752i
\(156\) 0 0
\(157\) −15.1814 5.52559i −1.21161 0.440990i −0.344348 0.938842i \(-0.611900\pi\)
−0.867262 + 0.497852i \(0.834122\pi\)
\(158\) 0 0
\(159\) −1.36502 2.36428i −0.108253 0.187500i
\(160\) 0 0
\(161\) 0.548323 0.199573i 0.0432139 0.0157286i
\(162\) 0 0
\(163\) −8.23541 + 6.91033i −0.645047 + 0.541259i −0.905564 0.424210i \(-0.860552\pi\)
0.260516 + 0.965469i \(0.416107\pi\)
\(164\) 0 0
\(165\) 10.5079 + 3.82455i 0.818037 + 0.297741i
\(166\) 0 0
\(167\) 0.554040 3.14212i 0.0428729 0.243145i −0.955839 0.293892i \(-0.905049\pi\)
0.998712 + 0.0507477i \(0.0161604\pi\)
\(168\) 0 0
\(169\) −9.57443 + 8.03390i −0.736495 + 0.617992i
\(170\) 0 0
\(171\) −1.70722 + 2.95700i −0.130555 + 0.226127i
\(172\) 0 0
\(173\) −2.22603 12.6245i −0.169242 0.959820i −0.944582 0.328276i \(-0.893532\pi\)
0.775340 0.631544i \(-0.217579\pi\)
\(174\) 0 0
\(175\) 3.39912 5.88745i 0.256949 0.445049i
\(176\) 0 0
\(177\) 1.77409 + 3.07281i 0.133349 + 0.230967i
\(178\) 0 0
\(179\) −18.1558 −1.35703 −0.678515 0.734587i \(-0.737376\pi\)
−0.678515 + 0.734587i \(0.737376\pi\)
\(180\) 0 0
\(181\) −7.05470 + 2.56770i −0.524372 + 0.190856i −0.590624 0.806947i \(-0.701118\pi\)
0.0662516 + 0.997803i \(0.478896\pi\)
\(182\) 0 0
\(183\) −0.0564798 0.320313i −0.00417511 0.0236782i
\(184\) 0 0
\(185\) 6.29635 + 21.7863i 0.462917 + 1.60176i
\(186\) 0 0
\(187\) 1.18459 + 6.71816i 0.0866260 + 0.491281i
\(188\) 0 0
\(189\) 0.717813 0.261262i 0.0522132 0.0190040i
\(190\) 0 0
\(191\) −19.3334 −1.39892 −0.699458 0.714674i \(-0.746575\pi\)
−0.699458 + 0.714674i \(0.746575\pi\)
\(192\) 0 0
\(193\) 1.56615 + 2.71266i 0.112734 + 0.195261i 0.916872 0.399182i \(-0.130706\pi\)
−0.804138 + 0.594443i \(0.797372\pi\)
\(194\) 0 0
\(195\) 1.32006 2.28641i 0.0945314 0.163733i
\(196\) 0 0
\(197\) −0.374905 2.12619i −0.0267109 0.151485i 0.968535 0.248876i \(-0.0800613\pi\)
−0.995246 + 0.0973913i \(0.968950\pi\)
\(198\) 0 0
\(199\) −12.4258 + 21.5221i −0.880840 + 1.52566i −0.0304309 + 0.999537i \(0.509688\pi\)
−0.850409 + 0.526122i \(0.823645\pi\)
\(200\) 0 0
\(201\) 3.97888 3.33868i 0.280649 0.235492i
\(202\) 0 0
\(203\) 1.16446 6.60399i 0.0817291 0.463509i
\(204\) 0 0
\(205\) −17.2101 6.26396i −1.20200 0.437494i
\(206\) 0 0
\(207\) 0.585166 0.491013i 0.0406719 0.0341277i
\(208\) 0 0
\(209\) 9.62351 3.50267i 0.665672 0.242285i
\(210\) 0 0
\(211\) −9.75574 16.8974i −0.671613 1.16327i −0.977447 0.211182i \(-0.932269\pi\)
0.305834 0.952085i \(-0.401065\pi\)
\(212\) 0 0
\(213\) −7.91277 2.88001i −0.542174 0.197335i
\(214\) 0 0
\(215\) 4.36016 + 3.65861i 0.297361 + 0.249515i
\(216\) 0 0
\(217\) 2.78998 + 2.34107i 0.189396 + 0.158922i
\(218\) 0 0
\(219\) −1.61189 + 9.14151i −0.108922 + 0.617726i
\(220\) 0 0
\(221\) 1.61062 0.108342
\(222\) 0 0
\(223\) 2.00165 0.134040 0.0670201 0.997752i \(-0.478651\pi\)
0.0670201 + 0.997752i \(0.478651\pi\)
\(224\) 0 0
\(225\) 1.54540 8.76441i 0.103027 0.584294i
\(226\) 0 0
\(227\) −17.9784 15.0856i −1.19327 1.00127i −0.999797 0.0201630i \(-0.993581\pi\)
−0.193470 0.981106i \(-0.561974\pi\)
\(228\) 0 0
\(229\) −20.5538 17.2467i −1.35823 1.13969i −0.976524 0.215409i \(-0.930892\pi\)
−0.381708 0.924283i \(-0.624664\pi\)
\(230\) 0 0
\(231\) −2.15297 0.783618i −0.141655 0.0515583i
\(232\) 0 0
\(233\) 1.76831 + 3.06281i 0.115846 + 0.200651i 0.918118 0.396308i \(-0.129709\pi\)
−0.802272 + 0.596959i \(0.796375\pi\)
\(234\) 0 0
\(235\) 39.2609 14.2898i 2.56110 0.932163i
\(236\) 0 0
\(237\) 6.79079 5.69815i 0.441109 0.370134i
\(238\) 0 0
\(239\) 22.6083 + 8.22874i 1.46241 + 0.532273i 0.946028 0.324086i \(-0.105057\pi\)
0.516381 + 0.856359i \(0.327279\pi\)
\(240\) 0 0
\(241\) 1.49091 8.45535i 0.0960377 0.544657i −0.898387 0.439205i \(-0.855260\pi\)
0.994424 0.105451i \(-0.0336288\pi\)
\(242\) 0 0
\(243\) 0.766044 0.642788i 0.0491418 0.0412348i
\(244\) 0 0
\(245\) 11.9610 20.7171i 0.764162 1.32357i
\(246\) 0 0
\(247\) −0.419868 2.38119i −0.0267155 0.151511i
\(248\) 0 0
\(249\) −0.546833 + 0.947143i −0.0346541 + 0.0600227i
\(250\) 0 0
\(251\) 8.96326 + 15.5248i 0.565756 + 0.979918i 0.996979 + 0.0776725i \(0.0247489\pi\)
−0.431223 + 0.902245i \(0.641918\pi\)
\(252\) 0 0
\(253\) −2.29115 −0.144043
\(254\) 0 0
\(255\) 7.96817 2.90018i 0.498986 0.181616i
\(256\) 0 0
\(257\) 3.90197 + 22.1292i 0.243398 + 1.38038i 0.824184 + 0.566323i \(0.191634\pi\)
−0.580786 + 0.814057i \(0.697255\pi\)
\(258\) 0 0
\(259\) −1.29007 4.46382i −0.0801610 0.277368i
\(260\) 0 0
\(261\) −1.52440 8.64532i −0.0943582 0.535132i
\(262\) 0 0
\(263\) −17.6717 + 6.43198i −1.08969 + 0.396613i −0.823504 0.567310i \(-0.807984\pi\)
−0.266181 + 0.963923i \(0.585762\pi\)
\(264\) 0 0
\(265\) 10.1782 0.625240
\(266\) 0 0
\(267\) 0.0822155 + 0.142401i 0.00503151 + 0.00871482i
\(268\) 0 0
\(269\) 7.11607 12.3254i 0.433875 0.751493i −0.563329 0.826233i \(-0.690479\pi\)
0.997203 + 0.0747403i \(0.0238128\pi\)
\(270\) 0 0
\(271\) −0.659909 3.74253i −0.0400866 0.227343i 0.958182 0.286159i \(-0.0923785\pi\)
−0.998269 + 0.0588163i \(0.981267\pi\)
\(272\) 0 0
\(273\) −0.270469 + 0.468466i −0.0163695 + 0.0283529i
\(274\) 0 0
\(275\) −20.4481 + 17.1580i −1.23307 + 1.03467i
\(276\) 0 0
\(277\) 4.64697 26.3543i 0.279209 1.58347i −0.446056 0.895005i \(-0.647172\pi\)
0.725265 0.688470i \(-0.241717\pi\)
\(278\) 0 0
\(279\) 4.48031 + 1.63070i 0.268229 + 0.0976275i
\(280\) 0 0
\(281\) 10.2170 8.57305i 0.609493 0.511426i −0.284988 0.958531i \(-0.591990\pi\)
0.894481 + 0.447106i \(0.147545\pi\)
\(282\) 0 0
\(283\) 27.5810 10.0387i 1.63952 0.596737i 0.652566 0.757732i \(-0.273693\pi\)
0.986955 + 0.160995i \(0.0514703\pi\)
\(284\) 0 0
\(285\) −6.36490 11.0243i −0.377024 0.653025i
\(286\) 0 0
\(287\) 3.52620 + 1.28343i 0.208145 + 0.0757586i
\(288\) 0 0
\(289\) −9.06001 7.60225i −0.532942 0.447191i
\(290\) 0 0
\(291\) 6.75476 + 5.66792i 0.395971 + 0.332259i
\(292\) 0 0
\(293\) −3.88949 + 22.0584i −0.227226 + 1.28866i 0.631157 + 0.775655i \(0.282580\pi\)
−0.858383 + 0.513009i \(0.828531\pi\)
\(294\) 0 0
\(295\) −13.2284 −0.770187
\(296\) 0 0
\(297\) −2.99935 −0.174040
\(298\) 0 0
\(299\) −0.0939329 + 0.532720i −0.00543228 + 0.0308080i
\(300\) 0 0
\(301\) −0.893360 0.749618i −0.0514924 0.0432073i
\(302\) 0 0
\(303\) −13.8667 11.6355i −0.796621 0.668444i
\(304\) 0 0
\(305\) 1.13949 + 0.414740i 0.0652469 + 0.0237479i
\(306\) 0 0
\(307\) 7.46672 + 12.9327i 0.426148 + 0.738110i 0.996527 0.0832715i \(-0.0265369\pi\)
−0.570379 + 0.821382i \(0.693204\pi\)
\(308\) 0 0
\(309\) −1.70320 + 0.619913i −0.0968916 + 0.0352656i
\(310\) 0 0
\(311\) 2.04270 1.71403i 0.115831 0.0971936i −0.583033 0.812449i \(-0.698134\pi\)
0.698863 + 0.715255i \(0.253690\pi\)
\(312\) 0 0
\(313\) −1.16799 0.425112i −0.0660185 0.0240288i 0.308800 0.951127i \(-0.400073\pi\)
−0.374818 + 0.927098i \(0.622295\pi\)
\(314\) 0 0
\(315\) −0.494535 + 2.80465i −0.0278639 + 0.158024i
\(316\) 0 0
\(317\) 24.4484 20.5146i 1.37316 1.15222i 0.401490 0.915863i \(-0.368492\pi\)
0.971667 0.236353i \(-0.0759520\pi\)
\(318\) 0 0
\(319\) −13.1652 + 22.8028i −0.737109 + 1.27671i
\(320\) 0 0
\(321\) −0.774738 4.39376i −0.0432416 0.245236i
\(322\) 0 0
\(323\) 3.88295 6.72546i 0.216053 0.374215i
\(324\) 0 0
\(325\) 3.15111 + 5.45788i 0.174792 + 0.302749i
\(326\) 0 0
\(327\) −3.16649 −0.175107
\(328\) 0 0
\(329\) −8.04422 + 2.92786i −0.443492 + 0.161418i
\(330\) 0 0
\(331\) 4.28280 + 24.2890i 0.235404 + 1.33504i 0.841761 + 0.539850i \(0.181519\pi\)
−0.606358 + 0.795192i \(0.707370\pi\)
\(332\) 0 0
\(333\) −3.58562 4.91359i −0.196491 0.269263i
\(334\) 0 0
\(335\) 3.36263 + 19.0704i 0.183720 + 1.04193i
\(336\) 0 0
\(337\) 3.93026 1.43050i 0.214095 0.0779241i −0.232746 0.972537i \(-0.574771\pi\)
0.446841 + 0.894613i \(0.352549\pi\)
\(338\) 0 0
\(339\) −13.5728 −0.737171
\(340\) 0 0
\(341\) −7.15023 12.3846i −0.387207 0.670662i
\(342\) 0 0
\(343\) −5.12429 + 8.87554i −0.276686 + 0.479234i
\(344\) 0 0
\(345\) 0.494535 + 2.80465i 0.0266249 + 0.150997i
\(346\) 0 0
\(347\) −17.5188 + 30.3435i −0.940460 + 1.62892i −0.175865 + 0.984414i \(0.556272\pi\)
−0.764596 + 0.644510i \(0.777061\pi\)
\(348\) 0 0
\(349\) −6.63920 + 5.57095i −0.355388 + 0.298206i −0.802949 0.596047i \(-0.796737\pi\)
0.447561 + 0.894253i \(0.352293\pi\)
\(350\) 0 0
\(351\) −0.122968 + 0.697387i −0.00656355 + 0.0372237i
\(352\) 0 0
\(353\) 32.4695 + 11.8179i 1.72818 + 0.629005i 0.998498 0.0547863i \(-0.0174478\pi\)
0.729678 + 0.683791i \(0.239670\pi\)
\(354\) 0 0
\(355\) 24.0491 20.1796i 1.27639 1.07102i
\(356\) 0 0
\(357\) −1.63261 + 0.594221i −0.0864069 + 0.0314495i
\(358\) 0 0
\(359\) −9.99659 17.3146i −0.527600 0.913829i −0.999482 0.0321681i \(-0.989759\pi\)
0.471883 0.881661i \(-0.343575\pi\)
\(360\) 0 0
\(361\) 6.89881 + 2.51096i 0.363095 + 0.132156i
\(362\) 0 0
\(363\) −1.53506 1.28807i −0.0805698 0.0676061i
\(364\) 0 0
\(365\) −26.5107 22.2451i −1.38763 1.16436i
\(366\) 0 0
\(367\) 0.159093 0.902263i 0.00830461 0.0470978i −0.980374 0.197147i \(-0.936832\pi\)
0.988678 + 0.150050i \(0.0479434\pi\)
\(368\) 0 0
\(369\) 4.91242 0.255731
\(370\) 0 0
\(371\) −2.08542 −0.108270
\(372\) 0 0
\(373\) 2.93440 16.6418i 0.151937 0.861680i −0.809595 0.586989i \(-0.800313\pi\)
0.961532 0.274691i \(-0.0885756\pi\)
\(374\) 0 0
\(375\) 11.1372 + 9.34524i 0.575124 + 0.482586i
\(376\) 0 0
\(377\) 4.76218 + 3.99594i 0.245265 + 0.205801i
\(378\) 0 0
\(379\) −7.40320 2.69455i −0.380277 0.138410i 0.144807 0.989460i \(-0.453744\pi\)
−0.525084 + 0.851050i \(0.675966\pi\)
\(380\) 0 0
\(381\) 6.39444 + 11.0755i 0.327597 + 0.567415i
\(382\) 0 0
\(383\) 2.20827 0.803745i 0.112837 0.0410694i −0.284984 0.958532i \(-0.591988\pi\)
0.397822 + 0.917463i \(0.369766\pi\)
\(384\) 0 0
\(385\) 6.54347 5.49062i 0.333486 0.279828i
\(386\) 0 0
\(387\) −1.43461 0.522155i −0.0729252 0.0265426i
\(388\) 0 0
\(389\) 1.24312 7.05007i 0.0630286 0.357453i −0.936940 0.349491i \(-0.886355\pi\)
0.999968 0.00796177i \(-0.00253434\pi\)
\(390\) 0 0
\(391\) −1.33092 + 1.11677i −0.0673073 + 0.0564775i
\(392\) 0 0
\(393\) −9.23425 + 15.9942i −0.465806 + 0.806800i
\(394\) 0 0
\(395\) 5.73902 + 32.5476i 0.288761 + 1.63765i
\(396\) 0 0
\(397\) −7.58725 + 13.1415i −0.380793 + 0.659553i −0.991176 0.132554i \(-0.957682\pi\)
0.610383 + 0.792107i \(0.291016\pi\)
\(398\) 0 0
\(399\) 1.30411 + 2.25879i 0.0652873 + 0.113081i
\(400\) 0 0
\(401\) −9.32332 −0.465584 −0.232792 0.972527i \(-0.574786\pi\)
−0.232792 + 0.972527i \(0.574786\pi\)
\(402\) 0 0
\(403\) −3.17271 + 1.15477i −0.158044 + 0.0575233i
\(404\) 0 0
\(405\) 0.647398 + 3.67158i 0.0321695 + 0.182442i
\(406\) 0 0
\(407\) −1.23468 + 18.2025i −0.0612006 + 0.902266i
\(408\) 0 0
\(409\) −4.79905 27.2167i −0.237298 1.34578i −0.837721 0.546099i \(-0.816112\pi\)
0.600423 0.799682i \(-0.294999\pi\)
\(410\) 0 0
\(411\) 15.3822 5.59866i 0.758748 0.276162i
\(412\) 0 0
\(413\) 2.71038 0.133369
\(414\) 0 0
\(415\) −2.03871 3.53116i −0.100077 0.173338i
\(416\) 0 0
\(417\) 1.52800 2.64657i 0.0748263 0.129603i
\(418\) 0 0
\(419\) 0.840202 + 4.76502i 0.0410465 + 0.232787i 0.998429 0.0560378i \(-0.0178467\pi\)
−0.957382 + 0.288824i \(0.906736\pi\)
\(420\) 0 0
\(421\) 13.4999 23.3825i 0.657944 1.13959i −0.323203 0.946330i \(-0.604760\pi\)
0.981147 0.193263i \(-0.0619072\pi\)
\(422\) 0 0
\(423\) −8.58473 + 7.20344i −0.417404 + 0.350243i
\(424\) 0 0
\(425\) −3.51490 + 19.9340i −0.170498 + 0.966940i
\(426\) 0 0
\(427\) −0.233471 0.0849767i −0.0112985 0.00411231i
\(428\) 0 0
\(429\) 1.62706 1.36527i 0.0785552 0.0659157i
\(430\) 0 0
\(431\) 22.1316 8.05525i 1.06604 0.388008i 0.251347 0.967897i \(-0.419126\pi\)
0.814695 + 0.579889i \(0.196904\pi\)
\(432\) 0 0
\(433\) 11.4407 + 19.8159i 0.549806 + 0.952292i 0.998287 + 0.0584996i \(0.0186316\pi\)
−0.448482 + 0.893792i \(0.648035\pi\)
\(434\) 0 0
\(435\) 30.7551 + 11.1939i 1.47459 + 0.536708i
\(436\) 0 0
\(437\) 1.99802 + 1.67654i 0.0955782 + 0.0801996i
\(438\) 0 0
\(439\) 10.8385 + 9.09458i 0.517293 + 0.434061i 0.863687 0.504029i \(-0.168149\pi\)
−0.346394 + 0.938089i \(0.612594\pi\)
\(440\) 0 0
\(441\) −1.11421 + 6.31901i −0.0530577 + 0.300905i
\(442\) 0 0
\(443\) −1.36923 −0.0650539 −0.0325270 0.999471i \(-0.510355\pi\)
−0.0325270 + 0.999471i \(0.510355\pi\)
\(444\) 0 0
\(445\) −0.613035 −0.0290606
\(446\) 0 0
\(447\) −1.38786 + 7.87095i −0.0656436 + 0.372283i
\(448\) 0 0
\(449\) 4.86123 + 4.07905i 0.229415 + 0.192502i 0.750248 0.661156i \(-0.229934\pi\)
−0.520833 + 0.853659i \(0.674378\pi\)
\(450\) 0 0
\(451\) −11.2870 9.47090i −0.531483 0.445967i
\(452\) 0 0
\(453\) −16.0923 5.85713i −0.756083 0.275192i
\(454\) 0 0
\(455\) −1.00837 1.74654i −0.0472730 0.0818792i
\(456\) 0 0
\(457\) 9.97593 3.63094i 0.466655 0.169848i −0.0979818 0.995188i \(-0.531239\pi\)
0.564636 + 0.825340i \(0.309016\pi\)
\(458\) 0 0
\(459\) −1.74231 + 1.46197i −0.0813240 + 0.0682390i
\(460\) 0 0
\(461\) 39.4060 + 14.3426i 1.83532 + 0.668002i 0.991288 + 0.131712i \(0.0420475\pi\)
0.844034 + 0.536290i \(0.180175\pi\)
\(462\) 0 0
\(463\) −2.56861 + 14.5673i −0.119373 + 0.676999i 0.865118 + 0.501568i \(0.167243\pi\)
−0.984492 + 0.175431i \(0.943868\pi\)
\(464\) 0 0
\(465\) −13.6169 + 11.4259i −0.631468 + 0.529865i
\(466\) 0 0
\(467\) 4.96688 8.60288i 0.229840 0.398094i −0.727921 0.685661i \(-0.759513\pi\)
0.957760 + 0.287567i \(0.0928465\pi\)
\(468\) 0 0
\(469\) −0.688974 3.90736i −0.0318138 0.180425i
\(470\) 0 0
\(471\) −8.07787 + 13.9913i −0.372208 + 0.644684i
\(472\) 0 0
\(473\) 2.28952 + 3.96557i 0.105272 + 0.182337i
\(474\) 0 0
\(475\) 30.3872 1.39426
\(476\) 0 0
\(477\) −2.56539 + 0.933727i −0.117461 + 0.0427524i
\(478\) 0 0
\(479\) −4.59002 26.0313i −0.209723 1.18940i −0.889832 0.456289i \(-0.849178\pi\)
0.680108 0.733112i \(-0.261933\pi\)
\(480\) 0 0
\(481\) 4.18169 + 1.03335i 0.190669 + 0.0471166i
\(482\) 0 0
\(483\) −0.101326 0.574648i −0.00461049 0.0261474i
\(484\) 0 0
\(485\) −30.8918 + 11.2437i −1.40272 + 0.510550i
\(486\) 0 0
\(487\) −17.1652 −0.777831 −0.388916 0.921273i \(-0.627150\pi\)
−0.388916 + 0.921273i \(0.627150\pi\)
\(488\) 0 0
\(489\) 5.37528 + 9.31027i 0.243079 + 0.421025i
\(490\) 0 0
\(491\) −6.60661 + 11.4430i −0.298152 + 0.516414i −0.975713 0.219052i \(-0.929704\pi\)
0.677561 + 0.735466i \(0.263037\pi\)
\(492\) 0 0
\(493\) 3.46714 + 19.6631i 0.156152 + 0.885582i
\(494\) 0 0
\(495\) 5.59112 9.68411i 0.251302 0.435268i
\(496\) 0 0
\(497\) −4.92745 + 4.13462i −0.221026 + 0.185463i
\(498\) 0 0
\(499\) −0.823590 + 4.67081i −0.0368689 + 0.209094i −0.997677 0.0681198i \(-0.978300\pi\)
0.960808 + 0.277214i \(0.0894111\pi\)
\(500\) 0 0
\(501\) −2.99818 1.09125i −0.133949 0.0487533i
\(502\) 0 0
\(503\) 22.8884 19.2056i 1.02054 0.856337i 0.0308470 0.999524i \(-0.490180\pi\)
0.989696 + 0.143187i \(0.0457351\pi\)
\(504\) 0 0
\(505\) 63.4171 23.0819i 2.82202 1.02713i
\(506\) 0 0
\(507\) 6.24927 + 10.8240i 0.277540 + 0.480713i
\(508\) 0 0
\(509\) −20.4994 7.46116i −0.908618 0.330710i −0.154917 0.987927i \(-0.549511\pi\)
−0.753701 + 0.657217i \(0.771733\pi\)
\(510\) 0 0
\(511\) 5.43182 + 4.55784i 0.240290 + 0.201627i
\(512\) 0 0
\(513\) 2.61562 + 2.19476i 0.115482 + 0.0969012i
\(514\) 0 0
\(515\) 1.17341 6.65476i 0.0517068 0.293244i
\(516\) 0 0
\(517\) 33.6125 1.47827
\(518\) 0 0
\(519\) −12.8192 −0.562701
\(520\) 0 0
\(521\) 6.24343 35.4083i 0.273530 1.55126i −0.470064 0.882632i \(-0.655769\pi\)
0.743594 0.668632i \(-0.233120\pi\)
\(522\) 0 0
\(523\) −4.06882 3.41415i −0.177917 0.149290i 0.549480 0.835507i \(-0.314826\pi\)
−0.727397 + 0.686216i \(0.759270\pi\)
\(524\) 0 0
\(525\) −5.20775 4.36982i −0.227285 0.190715i
\(526\) 0 0
\(527\) −10.1901 3.70890i −0.443889 0.161562i
\(528\) 0 0
\(529\) 11.2082 + 19.4132i 0.487315 + 0.844054i
\(530\) 0 0
\(531\) 3.33420 1.21355i 0.144692 0.0526636i
\(532\) 0 0
\(533\) −2.66485 + 2.23607i −0.115427 + 0.0968550i
\(534\) 0 0
\(535\) 15.6305 + 5.68902i 0.675764 + 0.245958i
\(536\) 0 0
\(537\) −3.15272 + 17.8800i −0.136050 + 0.771579i
\(538\) 0 0
\(539\) 14.7428 12.3706i 0.635016 0.532842i
\(540\) 0 0
\(541\) 9.97603 17.2790i 0.428903 0.742882i −0.567873 0.823116i \(-0.692234\pi\)
0.996776 + 0.0802343i \(0.0255669\pi\)
\(542\) 0 0
\(543\) 1.30366 + 7.39340i 0.0559453 + 0.317281i
\(544\) 0 0
\(545\) 5.90268 10.2237i 0.252843 0.437937i
\(546\) 0 0
\(547\) 4.42616 + 7.66634i 0.189249 + 0.327789i 0.945000 0.327070i \(-0.106061\pi\)
−0.755751 + 0.654859i \(0.772728\pi\)
\(548\) 0 0
\(549\) −0.325254 −0.0138815
\(550\) 0 0
\(551\) 28.1667 10.2518i 1.19994 0.436743i
\(552\) 0 0
\(553\) −1.17588 6.66872i −0.0500033 0.283583i
\(554\) 0 0
\(555\) 22.5486 2.41755i 0.957136 0.102619i
\(556\) 0 0
\(557\) 1.39239 + 7.89661i 0.0589973 + 0.334590i 0.999993 0.00382916i \(-0.00121886\pi\)
−0.940995 + 0.338419i \(0.890108\pi\)
\(558\) 0 0
\(559\) 1.01591 0.369761i 0.0429684 0.0156392i
\(560\) 0 0
\(561\) 6.82180 0.288017
\(562\) 0 0
\(563\) −14.6161 25.3158i −0.615994 1.06693i −0.990209 0.139591i \(-0.955421\pi\)
0.374215 0.927342i \(-0.377912\pi\)
\(564\) 0 0
\(565\) 25.3011 43.8228i 1.06443 1.84364i
\(566\) 0 0
\(567\) −0.132646 0.752275i −0.00557063 0.0315926i
\(568\) 0 0
\(569\) −2.72444 + 4.71888i −0.114215 + 0.197826i −0.917466 0.397815i \(-0.869768\pi\)
0.803251 + 0.595641i \(0.203102\pi\)
\(570\) 0 0
\(571\) −14.6585 + 12.2999i −0.613438 + 0.514736i −0.895733 0.444592i \(-0.853349\pi\)
0.282295 + 0.959328i \(0.408904\pi\)
\(572\) 0 0
\(573\) −3.35721 + 19.0397i −0.140249 + 0.795394i
\(574\) 0 0
\(575\) −6.38825 2.32513i −0.266409 0.0969648i
\(576\) 0 0
\(577\) −16.8685 + 14.1544i −0.702245 + 0.589253i −0.922411 0.386209i \(-0.873784\pi\)
0.220166 + 0.975462i \(0.429340\pi\)
\(578\) 0 0
\(579\) 2.94341 1.07131i 0.122324 0.0445222i
\(580\) 0 0
\(581\) 0.417715 + 0.723504i 0.0173297 + 0.0300160i
\(582\) 0 0
\(583\) 7.69452 + 2.80058i 0.318675 + 0.115988i
\(584\) 0 0
\(585\) −2.02245 1.69704i −0.0836180 0.0701638i
\(586\) 0 0
\(587\) −13.2897 11.1514i −0.548523 0.460266i 0.325917 0.945398i \(-0.394327\pi\)
−0.874441 + 0.485133i \(0.838771\pi\)
\(588\) 0 0
\(589\) −2.82692 + 16.0322i −0.116481 + 0.660597i
\(590\) 0 0
\(591\) −2.15899 −0.0888091
\(592\) 0 0
\(593\) −43.2482 −1.77599 −0.887995 0.459853i \(-0.847902\pi\)
−0.887995 + 0.459853i \(0.847902\pi\)
\(594\) 0 0
\(595\) 1.12478 6.37895i 0.0461115 0.261512i
\(596\) 0 0
\(597\) 19.0374 + 15.9743i 0.779149 + 0.653783i
\(598\) 0 0
\(599\) 34.5475 + 28.9888i 1.41157 + 1.18445i 0.955677 + 0.294416i \(0.0951251\pi\)
0.455895 + 0.890034i \(0.349319\pi\)
\(600\) 0 0
\(601\) −23.1578 8.42877i −0.944628 0.343817i −0.176636 0.984276i \(-0.556522\pi\)
−0.767992 + 0.640460i \(0.778744\pi\)
\(602\) 0 0
\(603\) −2.59703 4.49819i −0.105759 0.183180i
\(604\) 0 0
\(605\) 7.02035 2.55520i 0.285418 0.103884i
\(606\) 0 0
\(607\) 22.4319 18.8226i 0.910484 0.763987i −0.0617272 0.998093i \(-0.519661\pi\)
0.972211 + 0.234106i \(0.0752164\pi\)
\(608\) 0 0
\(609\) −6.30145 2.29354i −0.255348 0.0929390i
\(610\) 0 0
\(611\) 1.37805 7.81531i 0.0557500 0.316174i
\(612\) 0 0
\(613\) 31.1127 26.1066i 1.25663 1.05444i 0.260596 0.965448i \(-0.416081\pi\)
0.996033 0.0889891i \(-0.0283636\pi\)
\(614\) 0 0
\(615\) −9.15730 + 15.8609i −0.369258 + 0.639573i
\(616\) 0 0
\(617\) 1.69647 + 9.62117i 0.0682974 + 0.387334i 0.999726 + 0.0234095i \(0.00745216\pi\)
−0.931429 + 0.363924i \(0.881437\pi\)
\(618\) 0 0
\(619\) −1.80947 + 3.13409i −0.0727287 + 0.125970i −0.900096 0.435691i \(-0.856504\pi\)
0.827368 + 0.561661i \(0.189837\pi\)
\(620\) 0 0
\(621\) −0.381940 0.661540i −0.0153267 0.0265467i
\(622\) 0 0
\(623\) 0.125606 0.00503228
\(624\) 0 0
\(625\) −9.11977 + 3.31932i −0.364791 + 0.132773i
\(626\) 0 0
\(627\) −1.77835 10.0855i −0.0710206 0.402778i
\(628\) 0 0
\(629\) 8.15522 + 11.1756i 0.325170 + 0.445599i
\(630\) 0 0
\(631\) −6.58826 37.3639i −0.262274 1.48743i −0.776686 0.629888i \(-0.783101\pi\)
0.514411 0.857544i \(-0.328010\pi\)
\(632\) 0 0
\(633\) −18.3348 + 6.67332i −0.728742 + 0.265241i
\(634\) 0 0
\(635\) −47.6797 −1.89211
\(636\) 0 0
\(637\) −2.27190 3.93505i −0.0900160 0.155912i
\(638\) 0 0
\(639\) −4.21030 + 7.29245i −0.166557 + 0.288485i
\(640\) 0 0
\(641\) −1.14617 6.50026i −0.0452710 0.256745i 0.953770 0.300539i \(-0.0971665\pi\)
−0.999041 + 0.0437941i \(0.986055\pi\)
\(642\) 0 0
\(643\) −10.7396 + 18.6015i −0.423527 + 0.733571i −0.996282 0.0861564i \(-0.972542\pi\)
0.572754 + 0.819727i \(0.305875\pi\)
\(644\) 0 0
\(645\) 4.36016 3.65861i 0.171681 0.144058i
\(646\) 0 0
\(647\) −7.49876 + 42.5276i −0.294807 + 1.67193i 0.373181 + 0.927759i \(0.378267\pi\)
−0.667988 + 0.744172i \(0.732844\pi\)
\(648\) 0 0
\(649\) −10.0004 3.63986i −0.392552 0.142877i
\(650\) 0 0
\(651\) 2.78998 2.34107i 0.109348 0.0917539i
\(652\) 0 0
\(653\) 19.5234 7.10595i 0.764011 0.278077i 0.0695219 0.997580i \(-0.477853\pi\)
0.694489 + 0.719503i \(0.255630\pi\)
\(654\) 0 0
\(655\) −34.4273 59.6298i −1.34519 2.32993i
\(656\) 0 0
\(657\) 8.72272 + 3.17481i 0.340306 + 0.123861i
\(658\) 0 0
\(659\) 37.2188 + 31.2303i 1.44984 + 1.21656i 0.932699 + 0.360657i \(0.117447\pi\)
0.517139 + 0.855902i \(0.326997\pi\)
\(660\) 0 0
\(661\) −26.0974 21.8983i −1.01507 0.851745i −0.0260702 0.999660i \(-0.508299\pi\)
−0.989000 + 0.147915i \(0.952744\pi\)
\(662\) 0 0
\(663\) 0.279682 1.58615i 0.0108619 0.0616011i
\(664\) 0 0
\(665\) −9.72404 −0.377082
\(666\) 0 0
\(667\) −6.70586 −0.259652
\(668\) 0 0
\(669\) 0.347582 1.97124i 0.0134383 0.0762125i
\(670\) 0 0
\(671\) 0.747316 + 0.627073i 0.0288498 + 0.0242079i
\(672\) 0 0
\(673\) −5.84930 4.90815i −0.225474 0.189195i 0.523052 0.852301i \(-0.324794\pi\)
−0.748526 + 0.663106i \(0.769238\pi\)
\(674\) 0 0
\(675\) −8.36290 3.04385i −0.321888 0.117158i
\(676\) 0 0
\(677\) 6.86008 + 11.8820i 0.263654 + 0.456663i 0.967210 0.253977i \(-0.0817389\pi\)
−0.703556 + 0.710640i \(0.748406\pi\)
\(678\) 0 0
\(679\) 6.32947 2.30374i 0.242903 0.0884093i
\(680\) 0 0
\(681\) −17.9784 + 15.0856i −0.688933 + 0.578083i
\(682\) 0 0
\(683\) −14.3115 5.20895i −0.547614 0.199315i 0.0533723 0.998575i \(-0.483003\pi\)
−0.600986 + 0.799260i \(0.705225\pi\)
\(684\) 0 0
\(685\) −10.5975 + 60.1015i −0.404910 + 2.29636i
\(686\) 0 0
\(687\) −20.5538 + 17.2467i −0.784176 + 0.658001i
\(688\) 0 0
\(689\) 0.966630 1.67425i 0.0368257 0.0637840i
\(690\) 0 0
\(691\) −8.03276 45.5560i −0.305581 1.73303i −0.620759 0.784001i \(-0.713176\pi\)
0.315179 0.949032i \(-0.397935\pi\)
\(692\) 0 0
\(693\) −1.14557 + 1.98419i −0.0435167 + 0.0753732i
\(694\) 0 0
\(695\) 5.69670 + 9.86698i 0.216088 + 0.374276i
\(696\) 0 0
\(697\) −11.1729 −0.423205
\(698\) 0 0
\(699\) 3.32334 1.20960i 0.125700 0.0457512i
\(700\) 0 0
\(701\) −0.928540 5.26601i −0.0350705 0.198895i 0.962238 0.272208i \(-0.0877538\pi\)
−0.997309 + 0.0733133i \(0.976643\pi\)
\(702\) 0 0
\(703\) 14.3963 14.9702i 0.542968 0.564613i
\(704\) 0 0
\(705\) −7.25512 41.1458i −0.273243 1.54964i
\(706\) 0 0
\(707\) −12.9936 + 4.72929i −0.488675 + 0.177863i
\(708\) 0 0
\(709\) −13.7566 −0.516641 −0.258321 0.966059i \(-0.583169\pi\)
−0.258321 + 0.966059i \(0.583169\pi\)
\(710\) 0 0
\(711\) −4.43237 7.67709i −0.166227 0.287913i
\(712\) 0 0
\(713\) 1.82103 3.15412i 0.0681982 0.118123i
\(714\) 0 0
\(715\) 1.37506 + 7.79835i 0.0514243 + 0.291642i
\(716\) 0 0
\(717\) 12.0296 20.8359i 0.449254 0.778131i
\(718\) 0 0
\(719\) −16.4312 + 13.7874i −0.612780 + 0.514183i −0.895525 0.445012i \(-0.853199\pi\)
0.282745 + 0.959195i \(0.408755\pi\)
\(720\) 0 0
\(721\) −0.240422 + 1.36350i −0.00895380 + 0.0507795i
\(722\) 0 0
\(723\) −8.06800 2.93651i −0.300052 0.109210i
\(724\) 0 0
\(725\) −59.8487 + 50.2190i −2.22272 + 1.86509i
\(726\) 0 0
\(727\) −37.2519 + 13.5586i −1.38160 + 0.502860i −0.922662 0.385610i \(-0.873991\pi\)
−0.458934 + 0.888470i \(0.651769\pi\)
\(728\) 0 0
\(729\) −0.500000 0.866025i −0.0185185 0.0320750i
\(730\) 0 0
\(731\) 3.26291 + 1.18760i 0.120683 + 0.0439250i
\(732\) 0 0
\(733\) −10.2938 8.63754i −0.380211 0.319035i 0.432574 0.901598i \(-0.357605\pi\)
−0.812785 + 0.582564i \(0.802050\pi\)
\(734\) 0 0
\(735\) −18.3254 15.3768i −0.675941 0.567182i
\(736\) 0 0
\(737\) −2.70523 + 15.3421i −0.0996486 + 0.565135i
\(738\) 0 0
\(739\) 9.69479 0.356629 0.178314 0.983974i \(-0.442936\pi\)
0.178314 + 0.983974i \(0.442936\pi\)
\(740\) 0 0
\(741\) −2.41792 −0.0888246
\(742\) 0 0
\(743\) −4.07447 + 23.1075i −0.149478 + 0.847730i 0.814184 + 0.580606i \(0.197184\pi\)
−0.963662 + 0.267124i \(0.913927\pi\)
\(744\) 0 0
\(745\) −22.8261 19.1534i −0.836283 0.701724i
\(746\) 0 0
\(747\) 0.837797 + 0.702995i 0.0306534 + 0.0257212i
\(748\) 0 0
\(749\) −3.20255 1.16563i −0.117019 0.0425913i
\(750\) 0 0
\(751\) 9.75762 + 16.9007i 0.356061 + 0.616715i 0.987299 0.158874i \(-0.0507864\pi\)
−0.631238 + 0.775589i \(0.717453\pi\)
\(752\) 0 0
\(753\) 16.8454 6.13123i 0.613881 0.223434i
\(754\) 0 0
\(755\) 48.9089 41.0395i 1.77998 1.49358i
\(756\) 0 0
\(757\) −4.83714 1.76058i −0.175809 0.0639892i 0.252616 0.967567i \(-0.418709\pi\)
−0.428425 + 0.903577i \(0.640931\pi\)
\(758\) 0 0
\(759\) −0.397853 + 2.25634i −0.0144412 + 0.0818999i
\(760\) 0 0
\(761\) −34.1514 + 28.6564i −1.23799 + 1.03880i −0.240309 + 0.970696i \(0.577249\pi\)
−0.997679 + 0.0680989i \(0.978307\pi\)
\(762\) 0 0
\(763\) −1.20941 + 2.09476i −0.0437836 + 0.0758353i
\(764\) 0 0
\(765\) −1.47246 8.35072i −0.0532368 0.301921i
\(766\) 0 0
\(767\) −1.25631 + 2.17600i −0.0453628 + 0.0785707i
\(768\) 0 0
\(769\) 1.08877 + 1.88581i 0.0392622 + 0.0680042i 0.884989 0.465612i \(-0.154166\pi\)
−0.845727 + 0.533617i \(0.820833\pi\)
\(770\) 0 0
\(771\) 22.4705 0.809257
\(772\) 0 0
\(773\) −8.52475 + 3.10275i −0.306614 + 0.111598i −0.490745 0.871303i \(-0.663275\pi\)
0.184131 + 0.982902i \(0.441053\pi\)
\(774\) 0 0
\(775\) −7.36824 41.7874i −0.264675 1.50105i
\(776\) 0 0
\(777\) −4.62002 + 0.495335i −0.165742 + 0.0177700i
\(778\) 0 0
\(779\) 2.91264 + 16.5184i 0.104356 + 0.591833i
\(780\) 0 0
\(781\) 23.7332 8.63818i 0.849241 0.309098i
\(782\) 0 0
\(783\) −8.77869 −0.313725
\(784\) 0 0
\(785\) −30.1160 52.1625i −1.07489 1.86176i
\(786\) 0 0
\(787\) −11.5744 + 20.0475i −0.412584 + 0.714616i −0.995171 0.0981518i \(-0.968707\pi\)
0.582588 + 0.812768i \(0.302040\pi\)
\(788\) 0 0
\(789\) 3.26560 + 18.5202i 0.116259 + 0.659335i
\(790\) 0 0
\(791\) −5.18398 + 8.97891i −0.184321 + 0.319253i
\(792\) 0 0
\(793\) 0.176441 0.148051i 0.00626559 0.00525746i
\(794\) 0 0
\(795\) 1.76742 10.0235i 0.0626839 0.355498i
\(796\) 0 0
\(797\) 33.2727 + 12.1103i 1.17858 + 0.428968i 0.855700 0.517472i \(-0.173127\pi\)
0.322879 + 0.946440i \(0.395349\pi\)
\(798\) 0 0
\(799\) 19.5253 16.3837i 0.690756 0.579613i
\(800\) 0 0
\(801\) 0.154515 0.0562387i 0.00545950 0.00198710i
\(802\) 0 0
\(803\) −13.9208 24.1115i −0.491254 0.850877i
\(804\) 0 0
\(805\) 2.04427 + 0.744052i 0.0720510 + 0.0262244i
\(806\) 0 0
\(807\) −10.9025 9.14824i −0.383785 0.322033i
\(808\) 0 0
\(809\) 4.77841 + 4.00957i 0.168000 + 0.140969i 0.722912 0.690941i \(-0.242803\pi\)
−0.554911 + 0.831909i \(0.687248\pi\)
\(810\) 0 0
\(811\) 1.74244 9.88186i 0.0611853 0.346999i −0.938811 0.344432i \(-0.888072\pi\)
0.999997 0.00256748i \(-0.000817255\pi\)
\(812\) 0 0
\(813\) −3.80027 −0.133281
\(814\) 0 0
\(815\) −40.0805 −1.40396
\(816\) 0 0
\(817\) 0.905186 5.13356i 0.0316684 0.179601i
\(818\) 0 0
\(819\) 0.414382 + 0.347708i 0.0144797 + 0.0121499i
\(820\) 0 0
\(821\) 8.98243 + 7.53716i 0.313489 + 0.263049i 0.785932 0.618313i \(-0.212183\pi\)
−0.472443 + 0.881361i \(0.656628\pi\)
\(822\) 0 0
\(823\) 44.1166 + 16.0571i 1.53781 + 0.559716i 0.965518 0.260335i \(-0.0838331\pi\)
0.572290 + 0.820052i \(0.306055\pi\)
\(824\) 0 0
\(825\) 13.3465 + 23.1169i 0.464667 + 0.804827i
\(826\) 0 0
\(827\) 26.6774 9.70977i 0.927663 0.337642i 0.166380 0.986062i \(-0.446792\pi\)
0.761283 + 0.648420i \(0.224570\pi\)
\(828\) 0 0
\(829\) 15.9001 13.3418i 0.552234 0.463380i −0.323463 0.946241i \(-0.604847\pi\)
0.875697 + 0.482861i \(0.160403\pi\)
\(830\) 0 0
\(831\) −25.1470 9.15274i −0.872338 0.317505i
\(832\) 0 0
\(833\) 2.53419 14.3721i 0.0878044 0.497964i
\(834\) 0 0
\(835\) 9.11227 7.64610i 0.315343 0.264604i
\(836\) 0 0
\(837\) 2.38392 4.12908i 0.0824005 0.142722i
\(838\) 0 0
\(839\) 2.35129 + 13.3348i 0.0811755 + 0.460369i 0.998117 + 0.0613463i \(0.0195394\pi\)
−0.916941 + 0.399023i \(0.869349\pi\)
\(840\) 0 0
\(841\) −24.0327 + 41.6258i −0.828712 + 1.43537i
\(842\) 0 0
\(843\) −6.66865 11.5504i −0.229681 0.397818i
\(844\) 0 0
\(845\) −46.5973 −1.60299
\(846\) 0 0
\(847\) −1.43841 + 0.523538i −0.0494243 + 0.0179890i
\(848\) 0 0
\(849\) −5.09677 28.9052i −0.174921 0.992024i
\(850\) 0 0
\(851\) −4.17199 + 2.04560i −0.143014 + 0.0701224i
\(852\) 0 0
\(853\) 1.65996 + 9.41412i 0.0568361 + 0.322333i 0.999949 0.0101404i \(-0.00322784\pi\)
−0.943113 + 0.332474i \(0.892117\pi\)
\(854\) 0 0
\(855\) −11.9621 + 4.35385i −0.409095 + 0.148898i
\(856\) 0 0
\(857\) 23.1848 0.791977 0.395988 0.918255i \(-0.370402\pi\)
0.395988 + 0.918255i \(0.370402\pi\)
\(858\) 0 0
\(859\) 5.41234 + 9.37444i 0.184667 + 0.319852i 0.943464 0.331475i \(-0.107546\pi\)
−0.758798 + 0.651327i \(0.774213\pi\)
\(860\) 0 0
\(861\) 1.87625 3.24976i 0.0639425 0.110752i
\(862\) 0 0
\(863\) 6.71512 + 38.0833i 0.228585 + 1.29637i 0.855711 + 0.517453i \(0.173120\pi\)
−0.627126 + 0.778918i \(0.715769\pi\)
\(864\) 0 0
\(865\) 23.8964 41.3898i 0.812503 1.40730i
\(866\) 0 0
\(867\) −9.06001 + 7.60225i −0.307694 + 0.258186i
\(868\) 0 0
\(869\) −4.61704 + 26.1846i −0.156622 + 0.888250i
\(870\) 0 0
\(871\) 3.45633 + 1.25800i 0.117113 + 0.0426257i
\(872\) 0 0
\(873\) 6.75476 5.66792i 0.228614 0.191830i
\(874\) 0 0
\(875\) 10.4360 3.79839i 0.352801 0.128409i
\(876\) 0 0
\(877\) 9.74892 + 16.8856i 0.329198 + 0.570187i 0.982353 0.187037i \(-0.0598885\pi\)
−0.653155 + 0.757224i \(0.726555\pi\)
\(878\) 0 0
\(879\) 21.0479 + 7.66079i 0.709927 + 0.258392i
\(880\) 0 0
\(881\) 27.4241 + 23.0116i 0.923943 + 0.775280i 0.974720 0.223430i \(-0.0717253\pi\)
−0.0507770 + 0.998710i \(0.516170\pi\)
\(882\) 0 0
\(883\) −2.06569 1.73332i −0.0695160 0.0583309i 0.607368 0.794421i \(-0.292226\pi\)
−0.676884 + 0.736090i \(0.736670\pi\)
\(884\) 0 0
\(885\) −2.29709 + 13.0274i −0.0772157 + 0.437912i
\(886\) 0 0
\(887\) 43.6571 1.46586 0.732931 0.680303i \(-0.238152\pi\)
0.732931 + 0.680303i \(0.238152\pi\)
\(888\) 0 0
\(889\) 9.76917 0.327647
\(890\) 0 0
\(891\) −0.520832 + 2.95379i −0.0174485 + 0.0989556i
\(892\) 0 0
\(893\) −29.3121 24.5958i −0.980892 0.823066i
\(894\) 0 0
\(895\) −51.8527 43.5096i −1.73324 1.45436i
\(896\) 0 0
\(897\) 0.508315 + 0.185012i 0.0169722 + 0.00617736i
\(898\) 0 0
\(899\) −20.9277 36.2479i −0.697979 1.20893i
\(900\) 0 0
\(901\) 5.83479 2.12369i 0.194385 0.0707504i
\(902\) 0 0
\(903\) −0.893360 + 0.749618i −0.0297292 + 0.0249457i
\(904\) 0 0
\(905\) −26.3015 9.57296i −0.874291 0.318216i
\(906\) 0 0
\(907\) −6.72048 + 38.1138i −0.223150 + 1.26555i 0.643040 + 0.765832i \(0.277673\pi\)
−0.866190 + 0.499714i \(0.833438\pi\)
\(908\) 0 0
\(909\) −13.8667 + 11.6355i −0.459929 + 0.385927i
\(910\) 0 0
\(911\) 23.3908 40.5140i 0.774971 1.34229i −0.159840 0.987143i \(-0.551098\pi\)
0.934811 0.355146i \(-0.115569\pi\)
\(912\) 0 0
\(913\) −0.569617 3.23046i −0.0188516 0.106913i
\(914\) 0 0
\(915\) 0.606309 1.05016i 0.0200440 0.0347172i
\(916\) 0 0
\(917\) 7.05386 + 12.2176i 0.232939 + 0.403462i
\(918\) 0 0
\(919\) 12.8141 0.422699 0.211350 0.977411i \(-0.432214\pi\)
0.211350 + 0.977411i \(0.432214\pi\)
\(920\) 0 0
\(921\) 14.0328 5.10754i 0.462398 0.168299i
\(922\) 0 0
\(923\) −1.03546 5.87241i −0.0340827 0.193293i
\(924\) 0 0
\(925\) −21.9151 + 49.4999i −0.720565 + 1.62755i
\(926\) 0 0
\(927\) 0.314738 + 1.78497i 0.0103374 + 0.0586261i
\(928\) 0 0
\(929\) −25.7372 + 9.36759i −0.844412 + 0.307341i −0.727760 0.685832i \(-0.759438\pi\)
−0.116652 + 0.993173i \(0.537216\pi\)
\(930\) 0 0
\(931\) −21.9087 −0.718030
\(932\) 0 0
\(933\) −1.33328 2.30930i −0.0436495 0.0756032i
\(934\) 0 0
\(935\) −12.7166 + 22.0258i −0.415877 + 0.720320i
\(936\) 0 0
\(937\) −7.01203 39.7672i −0.229073 1.29914i −0.854744 0.519050i \(-0.826286\pi\)
0.625671 0.780087i \(-0.284825\pi\)
\(938\) 0 0
\(939\) −0.621472 + 1.07642i −0.0202810 + 0.0351277i
\(940\) 0 0
\(941\) 43.4106 36.4258i 1.41514 1.18745i 0.461262 0.887264i \(-0.347397\pi\)
0.953882 0.300183i \(-0.0970477\pi\)
\(942\) 0 0
\(943\) 0.651615 3.69550i 0.0212195 0.120342i
\(944\) 0 0
\(945\) 2.67616 + 0.974043i 0.0870556 + 0.0316856i
\(946\) 0 0
\(947\) −31.3693 + 26.3219i −1.01936 + 0.855348i −0.989548 0.144207i \(-0.953937\pi\)
−0.0298170 + 0.999555i \(0.509492\pi\)
\(948\) 0 0
\(949\) −6.17695 + 2.24823i −0.200512 + 0.0729805i
\(950\) 0 0
\(951\) −15.9575 27.6393i −0.517459 0.896265i
\(952\) 0 0
\(953\) 17.5298 + 6.38031i 0.567845 + 0.206679i 0.609957 0.792434i \(-0.291187\pi\)
−0.0421126 + 0.999113i \(0.513409\pi\)
\(954\) 0 0
\(955\) −55.2158 46.3316i −1.78674 1.49925i
\(956\) 0 0
\(957\) 20.1702 + 16.9248i 0.652011 + 0.547102i
\(958\) 0 0
\(959\) 2.17134 12.3143i 0.0701162 0.397649i
\(960\) 0 0
\(961\) −8.26762 −0.266697
\(962\) 0 0
\(963\) −4.46154 −0.143771
\(964\) 0 0
\(965\) −2.02785 + 11.5005i −0.0652788 + 0.370215i
\(966\) 0 0
\(967\) 6.72543 + 5.64331i 0.216275 + 0.181477i 0.744489 0.667635i \(-0.232693\pi\)
−0.528213 + 0.849112i \(0.677138\pi\)
\(968\) 0 0
\(969\) −5.94902 4.99182i −0.191110 0.160360i
\(970\) 0 0
\(971\) 27.0784 + 9.85575i 0.868989 + 0.316286i 0.737758 0.675066i \(-0.235885\pi\)
0.131231 + 0.991352i \(0.458107\pi\)
\(972\) 0 0
\(973\) −1.16721 2.02166i −0.0374189 0.0648115i
\(974\) 0 0
\(975\) 5.92215 2.15549i 0.189660 0.0690308i
\(976\) 0 0
\(977\) 19.2400 16.1443i 0.615543 0.516502i −0.280856 0.959750i \(-0.590618\pi\)
0.896399 + 0.443248i \(0.146174\pi\)
\(978\) 0 0
\(979\) −0.463444 0.168680i −0.0148117 0.00539103i
\(980\) 0 0
\(981\) −0.549855 + 3.11838i −0.0175555 + 0.0995623i
\(982\) 0 0
\(983\) 43.9027 36.8387i 1.40028 1.17497i 0.439302 0.898339i \(-0.355226\pi\)
0.960976 0.276633i \(-0.0892187\pi\)
\(984\) 0 0
\(985\) 4.02460 6.97081i 0.128234 0.222109i
\(986\) 0 0
\(987\) 1.48651 + 8.43042i 0.0473162 + 0.268343i
\(988\) 0 0
\(989\) −0.583099 + 1.00996i −0.0185415 + 0.0321148i
\(990\) 0 0
\(991\) −20.3498 35.2469i −0.646433 1.11966i −0.983968 0.178342i \(-0.942927\pi\)
0.337535 0.941313i \(-0.390407\pi\)
\(992\) 0 0
\(993\) 24.6637 0.782677
\(994\) 0 0
\(995\) −87.0644 + 31.6889i −2.76013 + 1.00460i
\(996\) 0 0
\(997\) 7.17373 + 40.6843i 0.227194 + 1.28848i 0.858445 + 0.512905i \(0.171431\pi\)
−0.631251 + 0.775579i \(0.717458\pi\)
\(998\) 0 0
\(999\) −5.46157 + 2.67791i −0.172797 + 0.0847254i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 888.2.bo.c.601.4 24
37.33 even 9 inner 888.2.bo.c.625.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bo.c.601.4 24 1.1 even 1 trivial
888.2.bo.c.625.4 yes 24 37.33 even 9 inner